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NASA NTRS1972

Resonant diffusion in the presence of strong plasma turbulence.

The diffusion equation which describes the evolution of the average one-particle distribution function for an ensemble of strongly turbulent plasmas is derived. The diffusion tensor is a time integral of the autocorrelation tensor of the fluctuations as observed by particles moving along statistically distributed orbits. These orbits contain the effects of fluctuations and thus differ from those encountered in weak turbulence theory. Two statistical orbit effects quadratic in the strength of the fluctuations affect the magnitude of the diffusion: (a) modification of the ensemble average orbits by the fluctuations, and (b) statistical dispersion in particle orbits about the average. The plasma trajectory equations are used to relate each to the diffusion tensor itself when the turbulence is electrostatic. The diffusion tensor is explicitly evaluated for a strongly turbulent unmagnetized plasma.

Birmingham, T. J.↗

NASA NTRS1986

Brans-Dicke-type models with nonmetricity

A scalar-tensor-nonmetric theory of gravity without matter fields is shown to be equivalent to the scalar-tensor-torsion theory proposed by German, the only difference being in the source of the Weyl vector or torsion vector (for equivalently the scalar field) which remains arbitrary in the theory. General extended conformal transformations, which leave the curvature tensor invariant, are presented and their implications for metric-affine theories and scalar-tensor theories are discussed.

Smalley, L. L.↗

NASA NTRS1988

Derivation of revised formulae for eddy viscous forces used in the ocean general circulation model

Presented is a re-derivation of the eddy viscous dissipation tensor commonly used in present oceanographic general circulation models. When isotropy is imposed, the currently-used form of the tensor fails to return to the laplacian operator. In this paper, the source of this error is identified in a consistent derivation of the tensor in both rectangular and earth spherical coordinates, and the correct form of the eddy viscous tensor is presented.

Chou, Ru Ling↗

NASA NTRS1990

A study of the topology of dissipating motions in direct numerical simulations of time-developing compressible and incompressible mixing layers

A preliminary investigation of the geometry of flow patterns in numerically simulated compressible and incompressible mixing layers was carried out using 3-D critical point methodology. Motions characterized by high rates of kinetic energy dissipation and/or high enstrophy were of particular interest. In the approach the partial derivatives of the velocity field are determined at every point in the flow. These are used to construct the invariants of the velocity gradient tensor and the rate-of-strain tensor (P, Q, R, and P(sub s), Q(sub s), R(sub s) respectively). For incompressible flow the first invariant is zero. For the conditions of the compressible simulation, the first invariant is found to be everywhere small, relative to the second and third invariants, and so in both cases the local topology at a point is mainly determined by the second and third invariants. The data at every grid point is used to construct scatter plots of Q versus R and Q(sub s) versus R(sub s). Most points map to a cluster near the origin in Q-R space. However, fine scale motions, that is motions which are characterized by velocity derivatives which scale with the square root of R(sub delta), tend to map to regions which lie far from the origin. Definite trends are observed for motions characterized by high enstrophy and/or high dissipation. The observed trends suggest that, for these motions, the second and third invariants of the velocity gradient and rate-of-strain tensors are strongly correlated. The second and third invariants of the rate-of-strain tensor are related by K(-Q(sub s))(exp 3/2), which is consistent with the above scaling of velocity derivatives. The quantity K appears to depend on Reynolds number with an upper limit K = 2(the square root of 3)/9 corresponding to locally axisymmetric flow. For both the compressible and incompressible mixing layer, regions corresponding to high rates of dissipation are found to be characterized by comparable magnitudes of R(sub ij)R(sub ij) and S(sub ij)S(sub ij). For the incompressible mixing layer, regions characterized by the highest values of enstrophy are found to have relatively low strain rates.

Chen, J. H.↗

NASA NTRS1994

Momentum transport near a magnetic X line in collisionless reconnection

Plasma dynamics and momentum transport near an X line during time-dependent magnetic reconnection in a collisionless plasma are investigated based on two-dimensional particle simulations. We find that a weakly skewed velocity distribution is formed near the magnetic X line, leading to the presence of off-diagonal elements of the plasma pressure tensor. Let the reconnection electric field be in the y direction. The gradients of the off-diagonal elements of the pressure tensor can provide a transport of the y momentum. During the normal magnetic reconnection, the momentum transport associated with the off-diagonal terms of the pressure tensor mediates a transfer of the y momentum from the region near the X line to regions outside the X line. A period of 'reverse magnetic reconnection,' during which the plasma kinetic energy is converted into magnetic energy, is also observed in the simulation. When reverse reconnection occurs, the gradients of the off-diagonal pressure tensor elements can mediate a transfer of y momentum into the X line. It is found that the inertial term also plays a significant role in the force balance near the magnetic X line. An explanation for the origin of the off-diagonal pressure terms is also given in this paper.

Cai, H. J.↗

NASA NTRS1993

Microwave Scattering Model for Grass Blade Structures

In this paper, the electromagnetic scattering solution for a grass blade with complex cross-section geometry is considered. It is assumed that the blade cross section is electrically small, but its length is large compared to the incident wavelength. In a recent study it has been shown that the scattering solution for such problems, in the form of a polarizability tensor, can be obtained using the low-frequency approximation in conjunction with the method of moments. In addition, the study shows that the relationship between the polarizability tensor of a dielectric cylinder and its dielectric constant can be approximated by a simple algebraic expression. The results of this study are used to show that this algebraic approximation is valid also for cylinders with cross sections the shape of grass blades, providing that proper values am selected for each of three constants appearing in the expression. These constants are dependent on cylinder shape, and if the relationship between the constants and the three parameters describing a grass blade shape can be determined, an algebraic approximation relating polarizability tensor to blade shape, as well as dielectric constant, can be formed. Since the elements of the polarizability tensor are dependent on only these parameters, this algebraic approximation can replace the cumbersome method of moments model. A conjugate gradient method is then implemented to correctly determine the three constants of the algebraic approximation for each blade shape. A third-order polynomial fit to the data is then determined for each constant, thus providing a complete analytic replacement to the numerical (moment method) scattering model. Comparisons of this approximation to the numerical model show an average error of less than 3%.

Stiles, James M.↗

NASA NTRS1997

Time Evolution of Modeled Reynolds Stresses in Planar Homogeneous Flows

The analytic expression of the time evolution of the Reynolds stress anisotropy tensor in all planar homogeneous flows is obtained by exact integration of the modeled differential Reynolds stress equations. The procedure is based on results of tensor representation theory, is applicable for general pressure-strain correlation tensors, and can account for any additional turbulence anisotropy effects included in the closure. An explicit solution of the resulting system of scalar ordinary differential equations is obtained for the case of a linear pressure-strain correlation tensor. The properties of this solution are discussed, and the dynamic behavior of the Reynolds stresses is studied, including limit cycles and sensitivity to initial anisotropies.

Jongen, T.↗

NASA NTRS1998

Multipoint Measurements in an Axisymmetric Sudden Expansion

A flying wire system was utilized in conjunction with a rake of nine cross-wire probes to obtain simultaneous velocity measurements in an axisymmetric sudden expansion at a Reynolds number of 41,000. From these measurements, the correlation tensor could be calculated. Knowledge of the two-point correlation tensor reveals more in-depth information of the physical attributes of this flow. The two point correlation tensor allowed for calculation of the integrated length scales in both the radial and axial directions. This gives insight into the growth of structures with increasing downstream distance and at different radial locations through out the sudden expansion. The length scales were calculated by integrating the two-point correlation tensor in the radial direction from the centerline to the outer pipe wall and by integrating between several step heights for the axial direction. Calculated correlations at z/h = 8 and 9 at r/R = 0.46 showed a correlation length of 1/3 step height for the radial direction. It was found that length scales in the radial direction became larger with increasing radius with peaks at 0.70 e r/R c 0.81 but then decreased slightly towards the wall. Length scales in the axial direction yielded a recirculating bubble on the order of 3 step heights in the recirculating region. After the recirculating region, the length scales decreased to 1/4 of a step height.

Eaton, Erroll L.↗

NASA NTRS2001

Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate, Epsilon(0) equations were considered. The emphasis of this paper is focused on the effects of the Epsilon(0)-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate (if change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters. Calculations show that the Epsilon(0)-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular Epsilon(0)-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the Epsilon(0)-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence.

Zhao, C. Y.↗

NASA NTRS1997

Mechanical signals in plant development: a new method for single cell studies

Cell division, which is critical to plant development and morphology, requires the orchestration of hundreds of intracellular processes. In the end, however, cells must make critical decisions, based on a discrete set of mechanical signals such as stress, strain, and shear, to divide in such a way that they will survive the mechanical loads generated by turgor pressure and cell enlargement within the growing tissues. Here we report on a method whereby tobacco protoplasts swirled into a 1.5% agarose entrapment medium will survive and divide. The application of a controlled mechanical load to agarose blocks containing protoplasts orients the primary division plane of the embedded cells. Photoelastic analysis of the agarose entrapment medium can identify the lines of principal stress within the agarose, confirming the hypothesis that cells divide either parallel or perpendicular to the principal stress tensors. The coincidence between the orientation of the new division wall and the orientation of the principal stress tensors suggests that the perception of mechanical stress is a characteristic of individual plant cells. The ability of a cell to determine a shear-free orientation for a new partition wall may be related to the applied load through the deformation of the matrix material. In an isotropic matrix a uniaxial load will produce a rotationally symmetric strain field, which will define a shear-free plane. Where high stress intensities combine with the loading geometry to produce multiaxial loads there will be no axis of rotational symmetry and hence no shear free plane. This suggests that two mechanisms may be orienting the division plane, one a mechanism that works in rotationally symmetrical fields, yielding divisions perpendicular to the compressive tensor, parallel to the long axis of the cell, and one in asymmetric fields, yielding divisions parallel to the short axis of the cell and the compressive tensor.

NASA Discipline Number 40-50↗

NASA NTRS2003

Inertial sensor and method of use

The inertial sensor of the present invention utilizes a proof mass suspended from spring structures forming a nearly degenerate resonant structure into which a perturbation is introduced, causing a split in frequency of the two modes so that the mode shape become uniquely defined, and to the first order, remains orthogonal. The resonator is provided with a mass or inertia tensor with off-diagonal elements. These off-diagonal elements are large enough to change the mode shape of the two nearly degenerate modes from the original coordinate frame. The spring tensor is then provided with a compensating off-diagonal element, such that the mode shape is again defined in the original coordinate frame. The compensating off-diagonal element in the spring tensor is provided by a biasing voltage that softens certain elements in the spring tensor. Acceleration disturbs the compensation and the mode shape again changes from the original coordinate frame. By measuring the change in the mode shape, the acceleration is measured.

Gutierrez, Roman C.↗

NASA NTRS2004

Reducing Errors by Use of Redundancy in Gravity Measurements

A methodology for improving gravity-gradient measurement data exploits the constraints imposed upon the components of the gravity-gradient tensor by the conditions of integrability needed for reconstruction of the gravitational potential. These constraints are derived from the basic equation for the gravitational potential and from mathematical identities that apply to the gravitational potential and its partial derivatives with respect to spatial coordinates. Consider the gravitational potential in a Cartesian coordinate system {x1,x2,x3}. If one measures all the components of the gravity-gradient tensor at all points of interest within a region of space in which one seeks to characterize the gravitational field, one obtains redundant information. One could utilize the constraints to select a minimum (that is, nonredundant) set of measurements from which the gravitational potential could be reconstructed. Alternatively, one could exploit the redundancy to reduce errors from noisy measurements. A convenient example is that of the selection of a minimum set of measurements to characterize the gravitational field at n3 points (where n is an integer) in a cube. Without the benefit of such a selection, it would be necessary to make 9n3 measurements because the gravitygradient tensor has 9 components at each point. The problem of utilizing the redundancy to reduce errors in noisy measurements is an optimization problem: Given a set of noisy values of the components of the gravity-gradient tensor at the measurement points, one seeks a set of corrected values - a set that is optimum in that it minimizes some measure of error (e.g., the sum of squares of the differences between the corrected and noisy measurement values) while taking account of the fact that the constraints must apply to the exact values. The problem as thus posed leads to a vector equation that can be solved to obtain the corrected values.

Kulikov, Igor↗

DOE OSTI2022

Characterization of anisotropic mechanical properties of polymer composites from a hybrid additive manufacturing-compression molding process using x-ray computer tomography

In this study, anisotropic stiffness tensors were reconstructed based on fiber orientation distributions obtained from X-ray computer tomography (xCT). A preform was manufactured via a big area additive manufacturing (BAAM) system with carbon fiber (CF) filled acrylonitrile butadiene styrene (ABS). The tailored preform from additive manufacturing (AM) was used in the compression molding (CM) process to produce a low-void high-performance thermoplastic composite panel. An xCT technique was employed to detect the fiber orientations in CF/ABS composites manufactured via three different methods: AM from BAAM, extrusion compression molding (ECM), and AM-CM. The anisotropic stiffness tensor was obtained from the composite panel manufactured via the three manufacturing methods (AM, ECM, and AMCM). A micromechanics theory was used to obtain the orthotropic stiffness tensors of the composite panels and compared with the experimental values. The predicted stiffness tensors of AM and AM-CM composite panels were used to study the deformation characteristics of a steering wheel during airbag deployment by performing finite element analysis (FEA). The approach developed in this study can be utilized for evaluating high-performance composites.

Pokkalla, Deepak↗

DOE OSTI2023

Zero-truncated Poisson regression for sparse multiway count data corrupted by false zeros

Abstract We propose a novel statistical inference methodology for multiway count data that is corrupted by false zeros that are indistinguishable from true zero counts. Our approach consists of zero-truncating the Poisson distribution to neglect all zero values. This simple truncated approach dispenses with the need to distinguish between true and false zero counts and reduces the amount of data to be processed. Inference is accomplished via tensor completion that imposes low-rank tensor structure on the Poisson parameter space. Our main result shows that an $N$-way rank-$R$ parametric tensor $\boldsymbol{\mathscr{M}}\in (0,\infty )^{I\times \cdots \times I}$ generating Poisson observations can be accurately estimated by zero-truncated Poisson regression from approximately $IR^2\log _2^2(I)$ non-zero counts under the nonnegative canonical polyadic decomposition. Our result also quantifies the error made by zero-truncating the Poisson distribution when the parameter is uniformly bounded from below. Therefore, under a low-rank multiparameter model, we propose an implementable approach guaranteed to achieve accurate regression in under-determined scenarios with substantial corruption by false zeros. Several numerical experiments are presented to explore the theoretical results.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

SBIR Phase I Final Report, TACO: Distributed and Heterogeneous Sparse Compiler

Tensor algebra is a powerful tool for computing, but writing optimized codes that operate on sparse tensors can be very complex. This project enables a Tensor Algebra Compiler (TACO) that simplifies this task from man-years to man-days and extends TACO to support complex and large distributed systems. This report details the hypotheses, approaches used, and findings in this project.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Comparative analysis of plasticity-based GND density estimation methods in crystal plasticity finite element models

In crystal plasticity finite element (CPFE) simulations, accurately quantifying geometrically necessary dislocations (GNDs) is critical for capturing strain gradients in polycrystals. We compare different methods for quantifying GNDs, all of which originate from the Nye tensor, which is computed as the curl of the plastic deformation gradient. The projection technique directly decomposes the Nye tensor onto individual screw and edge dislocation components to compute GNDs. This approach requires converting a nine-component Nye tensor into densities for a larger number of dislocation systems, a fundamentally underdetermined (non-unique) process, which is resolved using L2 minimization. In contrast, when employing CPFE analysis, one could directly compute dislocation densities on each slip system using shear gradients. Projection and slip gradient methods are compared with respect to their prediction of GNDs with changing grain size, strain, and grain neighborhoods, including multigrain junctions. Although these techniques match analytical GND densities for single slip, single crystal deformation, and are consistent with anticipated overall GND trends, we find that the GND densities from projection techniques are significantly lower than those predicted from CPFE-based slip gradients in polycrystals. A suggested improvement of only using the active dislocation systems in the projection technique almost entirely resolved this mismatch.

Crystal plasticity↗

DOE OSTI2024

Parallel Randomized Tucker Decomposition Algorithms

The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. Here, we propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data.

Tucker decompositions↗

DOE OSTI2022

High-Precision Characterization of Seismicity from the 2022 Hunga Tonga-Hunga Ha'apai Volcanic Eruption

The earthquake swarm accompanying the January 2022 Hunga Tonga-Hunga Ha'apai (HTHH) volcanic eruption includes a large number of posteruptive moderate-magnitude seismic events and presents a unique opportunity to use remote monitoring methods to characterize and compare seismic activity with other historical caldera-forming eruptions. We compute improved epicentroid locations, magnitudes, and regional moment tensors of seismic events from this earthquake swarm using regional to teleseismic surface-wave cross correlation and waveform modeling. Precise relative locations of 91 seismic events derived from 59,047 intermediate-period Rayleigh- and Love-wave cross-correlation measurements collapse into a small area surrounding the volcano and exhibit a southeastern time-dependent migration. Regional moment tensors and observed waveforms indicate that these events have a similar mechanism and exhibit a strong positive compensated linear vector dipole component. Precise relative magnitudes agree with regional moment tensor moment magnitude ($M_w$) estimates while also showing that event sizes and frequency increase during the days after the eruption followed by a period of several weeks of less frequent seismicity of a similar size. The combined information from visual observation and early geologic models indicate that the observed seismicity may be the result of a complex series of events that occurred after the explosive eruption on 15 January, possibly involving rapid resupply of the magma chamber shortly after the eruption and additional faulting and instability in the following weeks. In addition, we identify and characterize an $M_w$ 4.5 event five days before the paroxysmal explosion on 15 January, indicating that additional seismic events preceding the main eruption could have been identified with improved local monitoring. As a result, our analysis of the HTHH eruption sequence demonstrates the value of potentially utilizing teleseismic surface-wave cross correlation and waveform modeling methods to assist in the detailed analysis of remote volcanic eruption sequences.

58 GEOSCIENCES↗